Optimizing and Factorizing the Wilson Matrix Higham, Nicholas J. and Lettington, Matthew C. 2021 MIMS EPrint: 2021.3 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Mathematical Assoc. of America American Mathematical Monthly 121:1 February 18, 2021 3:37 p.m. paper.tex page 1 Optimizing and Factorizing the Wilson Matrix Nicholas J. Higham and Matthew C. Lettington Abstract. The Wilson matrix, W , is a 4 × 4 unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940s, owing to its mild ill-conditioning. We ask how close W is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations W = ZTZ with Z having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that W has an integer factor Z and two rational factors, up to signed permutations. This little 4 × 4 matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results. 1. INTRODUCTION. In the early days of digital computing there was much interest in constructing matrices that could be used to test methods for solving linear systems and computing eigenvalues. Such matrices should have known inverse or eigenvalues, preferably of a simple form. A famous example is the Hilbert matrix, with (i; j) el- ement 1=(i + j − 1), which was the subject of much investigation and about which a great deal is known [4], [15, Sec. 28.1]. This and other test matrices have been col- lected in books [11], [32, App. C] and made available in software, such as in MATLAB [13, Sec. 5.1] and Julia [35]. While the Hilbert matrix is defined for any dimension, some matrices of a specific small dimension have been proposed. Among these is the Wilson matrix 2 5 7 6 5 3 7 10 8 7 W = 6 7 ; 4 6 8 10 9 5 5 7 9 10 which was a favorite of John Todd [29, 30, 31] and has been used by various authors, for example in [2, 7, 9, 10, 14, 18]. The earliest appearance we know of the Wilson matrix is in a 1946 paper by Morris [24], who investigates a linear system containing the matrix “devised by Mr. T. S. Wilson.” A 1948 report by T. S. Wilson [33] acknowl- edges “Capt. J. Morris of the Royal Aircraft Establishment,” so it appears that this is the same T. S. Wilson—an employee of D. Napier and Son, a British engineering company developing aircraft engines at the time (see also [34]). Matrices with integer entries are of particular interest as test matrices because they are stored exactly in floating-point arithmetic, provided that the entries are not too large. By contrast, the Hilbert matrix is not stored exactly, and this can lead to diffi- culties in interpreting the results of computational experiments, as explained by Moler [21]. The Wilson matrix is symmetric positive definite and has determinant det(W ) = 1 OPTIMIZING THE WILSON MATRIX 1 Mathematical Assoc. of America American Mathematical Monthly 121:1 February 18, 2021 3:37 p.m. paper.tex page 2 (i.e., is unimodular) and inverse 2 68 −41 −17 103 −41 25 10 −6 W −1 = 6 7 : 4−17 10 5 −35 10 −6 −3 2 The Wilson matrix is mildly ill-conditioned, with 2-norm condition number κ2(W ) = −1 3 kW k2kW k2 ≈ 2:98409 × 10 , where kAk2 = maxx6=0 kAxk2=kxk2 and kxk2 = T 1=2 (x x) . Recall that κ2(A) ≥ 1 and that κ2(A) is a measure of the sensitivity of a linear system Ax = b to perturbations in A and b. Matrices with a large condi- tion number are of interest for test purposes as they can pose various difficulties for methods for solving linear systems and other problems. We do not know how Wilson, working in the pre-digital computer era, constructed his matrix, and in particular to what extent he maximized the condition number subject to the matrix entries being small integers. Moler [22] asked how ill-conditioned W is relative to matrices in the set 4×4 S = f A 2 R : A is nonsingular and symmetric with integer entries between 1 and 10 g: (1) He carried out an experiment in which he generated one million random matrices from S. About 0.21 percent of the matrices had a larger condition number than that of W . The matrix with the largest condition number was 2 1 3 10 10 3 3 4 8 9 A = 6 7 ; (2) 1 4 10 8 3 9 5 10 9 9 3 4 which is not positive definite and has κ2(A1) ≈ 4:80867 × 10 , determinant 1, and inverse 2 573 −804 159 253 −804 1128 −223 −35 A−1 = 6 7 : 1 4 159 −223 44 75 25 −35 7 1 We will investigate the questions of what are the most ill-conditioned matrices in S and what are the most ill-conditioned positive definite matrices in P, the subset of S of positive definite matrices: 4×4 P = f A 2 R : A is symmetric positive definite with integer entries between 1 and 10 g: (3) We begin, in Section 2, by obtaining upper bounds on the condition numbers for these two cases. In Section 3 we determine the maximal condition numbers experimentally, by an exhaustive search. Wilson may have constructed W as the product ZTZ, where Z is a simpler integral matrix (one with integer entries). In Section 4 we identify a block triangular integral factor Z. By exploiting recent research that links the existence of these factorizations 2 Mathematical Assoc. of America American Mathematical Monthly 121:1 February 18, 2021 3:37 p.m. paper.tex page 3 to number-theoretic considerations of quadratic forms, we show that W has not only an integral factor Z but also two rational factors, up to signed permutations. Using this new theory we also identify an integer factor and two rational factors of the most ill-conditioned matrix in P. The Wilson matrix may only be 4 × 4, but it raises some interesting challenges. This should not be surprising because Taussky noted in 1961 that “matrices with integral elements have been studied for a very long time and an enormous number of problems arise, both theoretical and practical.” [27] Our study is an example of work on what have recently been termed “Bohemian matrices,” defined as families of matrices whose entries are drawn from a finite discrete set, typically made up of small integers [6]. For other recent results on this topic, see [3, 8]. 2. CONDITION NUMBER BOUNDS. We wish to obtain upper bounds on κ2(A) −1 for A 2 S, where S is defined in (1). We therefore need to bound kAk2 and kA k2. First we consider kAk2. We will use the inequality kAk2 ≤ kAkF , where the Frobe- P 2 1=2 nius norm is given by kAkF = i;j aij . Equality in this inequality holds only for the zero matrix and rank-1 matrices, so since A 2 S is nonsingular we have strict inequality. Nonsingularity also implies that matrices in S must have at least three en- tries not equal to 10, and they include, for example, 2 10 10 10 10 3 2 10 10 10 10 3 10 9 10 10 10 9 10 10 6 7 ; 6 7 : 4 10 10 9 10 5 4 10 10 10 0 5 10 10 10 9 10 10 9 10 (Both of these matrices have 2-norm condition number approximately 1:5 × 102, so they are quite well-conditioned.) Hence p 1=2 A 2 S ) kAk2 < kAkF ≤ (13 × 100 + 3 × 81) = 1543: (4) −1 Bound 1. We now derive a bound on kA k2 from first principles. The inverse of n×n −1 A 2 R is given by A = adj(A)= det(A), where the adjugate matrix i+j adj(A) = (−1) det(Aji); with Aij denoting the submatrix of A obtained by deleting row i and column j. Since j det(A)j ≥ 1 for A 2 S, jA−1j ≤ j adj(A)j; (5) where the absolute value and the inequality are taken componentwise. We note in n×n passing that a nonsingular integral matrix A 2 R has an integral inverse if and only if det(A) = ±1 [1, Theorem 2]. The “if” is obvious, but the “only if” is nontrivial. Hence if A and A−1 have integer entries then (5) is an equality. Since A 2 S is nonsingular, every 3 × 3 submatrix of A 2 S must contain at least one entry less than or equal to 9. Hence from (5) we have, for A 2 S, 2 p jA−1 j ≤ j det(A )j < 10 × 31=2 × 281; ij ji (6) where we have used Hadamard’s inequality [17, Cor. 7.8.3], which states that for B 2 n×n Qn R , j det(B)j ≤ k=1 kbkk2, where bk is the kth column of B.